In this paper we will consider solitary waves to an asymptotic water wave model derived by
Bona–Chen–Saut [1]} (generalized to include the surface tension in [2]} and in higher dimensions Bona–Colin–Lannes [3]}) as an extended system of the classical Boussinesq equation. Specifically, it is a three-parameter family of Boussinesq systems for one dimensional surfaces that takes the following form
\(\left\lbrace \begin{array}{l}\eta _t + u_x + (u\eta )_x + a u_{xxx} - b \eta _{xxt} = 0, \\u_t + \eta _x + {12}(u^2)_x + c \eta _{xxx} - d u_{xxt} = 0,\end{array}\right.\) 
